## The Annals of Statistics

- Ann. Statist.
- Volume 17, Number 1 (1989), 252-267.

### Universal Domination and Stochastic Domination: $U$-Admissibility and $U$- Inadmissibility of the Least Squares Estimator

Lawrence D. Brown and Jiunn T. Hwang

#### Abstract

Assume the standard linear model $X_{n \times 1} = A_{n \times p} \theta_{p \times 1} + \varepsilon_{n \times 1},$ where $\varepsilon$ has an $n$-variate normal distribution with zero mean vector and identity covariance matrix. The least squares estimator for the coefficient $\theta$ is $\hat{\theta} \equiv (A'A)^{-1}A'X$. It is well known that $\hat{\theta}$ is dominated by James-Stein type estimators under the sum of squared error loss $|\theta - \hat{\theta}|^2$ when $p \geq 3$. In this article we discuss the possibility of improving upon $\hat{\theta}$, simultaneously under the "universal" class of losses: $\{L(|\theta - \hat{\theta}|): L(\cdot) \text{any nondecreasing function}\}.$ An estimator that can be so improved is called universally inadmissible ($U$-inadmissible). Otherwise it is called $U$-admissible. We prove that $\hat{\theta}$ is $U$-admissible for any $p$ when $A'A = I$. Furthermore, if $A'A \neq I$, then $\hat{\theta}$ is $U$-inadmissible if $p$ is "large enough." In a special case, $p \geq 4$ is large enough. The results are surprising. Implications are discussed.

#### Article information

**Source**

Ann. Statist., Volume 17, Number 1 (1989), 252-267.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347014

**Digital Object Identifier**

doi:10.1214/aos/1176347014

**Mathematical Reviews number (MathSciNet)**

MR981448

**Zentralblatt MATH identifier**

0674.62007

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62C05: General considerations

Secondary: 62F11 62J07: Ridge regression; shrinkage estimators

**Keywords**

Decision theory under a broad class of loss functions James-Stein positive part estimator admissibility

#### Citation

Brown, Lawrence D.; Hwang, Jiunn T. Universal Domination and Stochastic Domination: $U$-Admissibility and $U$- Inadmissibility of the Least Squares Estimator. Ann. Statist. 17 (1989), no. 1, 252--267. doi:10.1214/aos/1176347014. https://projecteuclid.org/euclid.aos/1176347014